Everything Maths Grade 11

11.2 CHAPTER 11. QUADRATIC FUNCTIONS ANDGRAPHS

Sketching Graphs of theFormf(x)=a(x+

p)

2

+q

EMBBB

In order to sketch graphs of the form f(x) = a(x + p)^2 + q, we need to determinefive characteristics:

1. sign of a


2. domain and range


3. turning point


4. y-intercept


5. x-intercept (if appropriate)

For example, sketch thegraph of g(x) =−^12 (x + 1)^2 − 3. Mark the intercepts, turning point and axis
of symmetry.

Firstly, we determine that a < 0. This means that the graph will have a maximalturning point.

The domain of the graphis{x : x∈R} because f(x) is defined for all x∈R. The range of the graph
is determined as follows:

(x + 1)^2 ≥ 0

−

1

2

(x + 1)^2 ≤ 0

−

1

2

(x + 1)^2 − 3 ≤− 3

∴ f(x)≤− 3

Therefore the range of the graph is{f(x) : f(x)∈ (−∞,− 3]}.

Using the fact that the maximum value that f(x) achieves is− 3 , then the y-coordinate of the turning
point is− 3. The x-coordinate is determined as follows:

−

1

2

(x + 1)^2 − 3 =− 3

−

1

2

(x + 1)^2 − 3 + 3 = 0

−

1

2

(x + 1)^2 = 0

Divide both sides by−^12 : (x + 1)^2 = 0

Take square root of bothsides: x + 1 = 0

∴ x =− 1

The coordinates of the turning point are: (−1;−3).

The y-intercept is obtained bysetting x = 0. This gives:

yint =−

1

2

(0 + 1)^2 − 3

=−

1

2

(1)− 3

=−

1

2

− 3

=−

1

2

− 3

=−

7

2